12.3 Multidisciplinary design optimization

Multidisciplinary design optimization (MDO) tackles complex systems by considering interacting disciplines simultaneously. This approach enables designers to explore wider design spaces and find innovative solutions that may be missed when optimizing disciplines separately.

MDO addresses challenges in traditional design approaches, which often lead to suboptimal performance and longer design cycles. By allowing simultaneous optimization of multiple disciplines, MDO identifies optimal trade-offs, reduces iterations, and improves overall system performance.

Multidisciplinary design optimization overview

• Multidisciplinary design optimization (MDO) is a field that addresses the challenges of designing complex systems with interacting disciplines
• MDO aims to find optimal designs by considering the coupling and trade-offs between different disciplines simultaneously
• Enables the exploration of a wider design space and the identification of innovative solutions that may not be apparent when disciplines are optimized separately

Challenges of traditional design approaches

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• Traditional design approaches often optimize each discipline separately, leading to suboptimal overall system performance
• Lack of communication and coordination between disciplines can result in design inconsistencies and rework
• Sequential design processes may not adequately capture the interactions and trade-offs between disciplines
• Can lead to longer design cycles and increased development costs

Benefits of multidisciplinary optimization

• Allows for the simultaneous optimization of multiple disciplines, considering their interactions and constraints
• Enables the identification of optimal trade-offs between conflicting objectives and constraints
• Facilitates the exploration of a larger design space, potentially leading to innovative and improved designs
• Reduces design iterations and rework by addressing disciplinary coupling early in the design process
• Can lead to shorter design cycles, reduced development costs, and improved system performance

Formulation of MDO problems

• MDO problems involve the optimization of a system with multiple interacting disciplines
• Require the definition of design variables, constraints, and objective functions that capture the system's performance goals
• Involve the mathematical formulation of the optimization problem, considering the coupling between disciplines

Design variables and constraints

• Design variables represent the parameters that can be modified to optimize the system (wing shape, material properties)
• Constraints define the feasible region of the design space and ensure the system meets performance and safety requirements (maximum stress, minimum lift)
• Design variables and constraints may be shared across multiple disciplines or specific to individual disciplines

Objective functions for optimization

• Objective functions quantify the performance goals of the system and guide the optimization process (minimize weight, maximize fuel efficiency)
• Can be single-objective (optimize one performance metric) or multi-objective (trade-off between multiple conflicting objectives)
• Often involve a combination of disciplinary objectives and system-level objectives

Coupling between disciplines

• Coupling refers to the interdependencies and interactions between different disciplines in a system
• Can be unidirectional (output of one discipline affects another) or bidirectional (disciplines influence each other)
• Coupling must be properly modeled and managed in MDO to ensure consistent and accurate system analysis and optimization
• Techniques such as coupling variables, coupling constraints, and iterative solution methods are used to handle disciplinary coupling

MDO architectures and strategies

• MDO architectures define the organizational structure and information flow between disciplines during the optimization process
• Different architectures have been developed to address the challenges of disciplinary coupling and computational efficiency
• The choice of MDO architecture depends on the problem characteristics, available resources, and desired level of disciplinary autonomy

Single vs multi-level optimization

• Single-level optimization treats the entire system as a single integrated optimization problem, with all design variables and constraints considered simultaneously
• Multi-level optimization decomposes the system into disciplinary subproblems, which are optimized separately and coordinated through a system-level optimizer
• Single-level optimization provides a more comprehensive view of the system but can be computationally expensive for large-scale problems
• Multi-level optimization allows for disciplinary autonomy and parallel computation but requires careful coordination and consistency management

Collaborative optimization approach

• Collaborative optimization is a multi-level MDO architecture that allows disciplines to optimize their own subproblems while satisfying system-level targets
• Disciplines are given autonomy to choose their own design variables and optimization methods
• System-level optimizer coordinates the disciplinary subproblems by setting targets and managing consistency constraints
• Enables disciplinary experts to focus on their domain-specific optimization while ensuring overall system compatibility

Concurrent subspace optimization

• Concurrent subspace optimization is an MDO architecture that divides the design space into subspaces, each optimized by a disciplinary subproblem
• Subspaces are optimized concurrently, with the system-level optimizer coordinating the sharing of design variables and constraints between subspaces
• Allows for the efficient exploration of the design space by focusing on the most influential design variables for each discipline
• Requires careful selection of subspace boundaries and coordination strategies to ensure consistency and optimality

Analytical target cascading

• Analytical target cascading is a hierarchical MDO architecture that propagates system-level targets down to disciplinary subproblems
• Targets are cascaded through multiple levels of the system hierarchy, with each level optimizing its local design variables to meet the assigned targets
• Enables the decomposition of complex systems into manageable subproblems while maintaining system-level consistency
• Requires the definition of appropriate targets and coordination strategies to ensure convergence and optimality

Sensitivity analysis in MDO

• Sensitivity analysis assesses how changes in design variables and parameters affect the system's performance and constraints
• Helps identify the most influential design variables and understand the robustness of the optimal solution
• Provides insights into the trade-offs and interactions between disciplines
• Informs decision-making and guides the allocation of resources for further analysis and optimization

Local vs global sensitivity analysis

• Local sensitivity analysis evaluates the impact of small perturbations around a specific design point
• Provides information about the local behavior of the system and the sensitivity of the optimal solution to small changes in design variables
• Global sensitivity analysis assesses the influence of design variables over the entire design space
• Captures the overall importance and interactions of design variables, considering their full range of variation
• Global sensitivity analysis is more comprehensive but computationally demanding compared to local sensitivity analysis

Sensitivity analysis methods

• Finite difference methods approximate sensitivities by perturbing design variables and evaluating the corresponding changes in system performance
• Adjoint methods efficiently compute sensitivities by solving an adjoint problem, which involves the transposed of the system's governing equations
• Automatic differentiation tools can generate exact sensitivities by applying chain rule differentiation to the computer code implementing the system analysis
• Sampling-based methods (Monte Carlo, Latin Hypercube) estimate sensitivities by evaluating the system at multiple design points and analyzing the statistical properties of the outputs

Surrogate modeling techniques

• Surrogate models, also known as metamodels, are simplified approximations of the complex disciplinary analyses in MDO
• Used to reduce the computational burden of repeated function evaluations during optimization and sensitivity analysis
• Surrogate models are constructed using a limited number of high-fidelity simulations and can be used to predict system performance at untested design points
• Various techniques are available for building accurate and efficient surrogate models

Response surface methodology

• Response surface methodology (RSM) fits a polynomial function to the input-output relationship of a system
• Commonly used polynomial functions include linear, quadratic, and cubic models
• RSM is simple to implement and provides a global approximation of the system behavior
• Limitations include the difficulty in capturing highly nonlinear or discontinuous responses

Kriging and Gaussian processes

• Kriging is a probabilistic surrogate modeling technique that models the system response as a realization of a Gaussian process
• Assumes that the system outputs are spatially correlated, with closer design points having more similar responses
• Provides both a prediction of the system performance and an estimate of the prediction uncertainty
• Kriging models can capture complex nonlinear behaviors and provide a measure of the model's confidence

Artificial neural networks

• Artificial neural networks (ANNs) are machine learning models inspired by the structure and function of biological neural networks
• ANNs consist of interconnected nodes (neurons) organized in layers, which process and transmit information
• Can learn complex nonlinear relationships between inputs and outputs through training on a dataset of high-fidelity simulations
• ANNs are flexible and can handle high-dimensional design spaces, but require careful architecture selection and training

Proper orthogonal decomposition

• Proper orthogonal decomposition (POD) is a model reduction technique that extracts the most important features or modes from a set of high-fidelity simulations
• POD modes represent the dominant patterns of variability in the system response and can be used to construct a reduced-order surrogate model
• POD-based surrogate models are computationally efficient and can capture the essential dynamics of the system
• Particularly useful for systems with high-dimensional output spaces, such as fluid flow or structural dynamics problems

Uncertainty quantification in MDO

• Uncertainty quantification (UQ) assesses the impact of uncertainties in design variables, parameters, and models on the system performance and reliability
• UQ is crucial in MDO to ensure the robustness and safety of the optimal design under real-world conditions
• Various techniques are employed to propagate uncertainties through the system analysis and optimize the design under uncertainty

Types of uncertainties

• Aleatory uncertainties arise from inherent randomness or variability in the system (material properties, manufacturing tolerances)
• Epistemic uncertainties result from a lack of knowledge or incomplete information about the system (model assumptions, parameter estimates)
• Aleatory uncertainties are typically modeled using probability distributions, while epistemic uncertainties may be represented using intervals or fuzzy sets

Propagation of uncertainties

• Uncertainty propagation methods compute the statistical properties of the system outputs given the uncertainties in the inputs
• Monte Carlo simulation is a widely used technique that generates random samples from the input distributions and evaluates the system response for each sample
• Polynomial chaos expansion represents the system response as a series expansion of orthogonal polynomials in the random input variables
• Stochastic collocation methods evaluate the system at specific collocation points in the input space and interpolate the response using polynomial basis functions

Reliability-based design optimization

• Reliability-based design optimization (RBDO) incorporates reliability constraints into the MDO formulation to ensure the system meets a target reliability level
• RBDO aims to find the optimal design that minimizes the objective function while satisfying the reliability constraints under uncertainty
• Reliability constraints are typically formulated using probability of failure or reliability index, which measure the likelihood of the system violating performance or safety requirements
• RBDO methods include first-order reliability method (FORM), second-order reliability method (SORM), and simulation-based approaches (Monte Carlo, importance sampling)

Applications of MDO in aerodynamics

• MDO has been extensively applied in the field of aerodynamics to optimize aircraft performance, efficiency, and environmental impact
• Aerodynamic design optimization involves the simultaneous consideration of multiple disciplines, such as aerodynamics, structures, propulsion, and controls
• MDO enables the exploration of novel aircraft configurations and the identification of optimal trade-offs between conflicting objectives

Aircraft wing design optimization

• Wing design optimization aims to find the optimal wing shape, planform, and structural properties to maximize aerodynamic efficiency and minimize weight
• MDO approaches consider the coupling between aerodynamics and structures, accounting for the aeroelastic effects and structural integrity constraints
• High-fidelity computational fluid dynamics (CFD) and finite element analysis (FEA) are often used to evaluate the aerodynamic and structural performance of the wing
• Surrogate modeling techniques are employed to reduce the computational burden of repeated high-fidelity simulations during optimization

Propulsion system integration

• Propulsion system integration optimization seeks to find the optimal placement, sizing, and configuration of engines on an aircraft
• MDO approaches consider the interactions between the propulsion system and other disciplines, such as aerodynamics, structures, and controls
• Objectives may include maximizing thrust, minimizing fuel consumption, and reducing noise and emissions
• High-fidelity CFD simulations are used to analyze the flow field around the engine and its impact on aircraft performance
• Surrogate models and reduced-order methods are employed to efficiently explore the design space and optimize the propulsion system integration

Aerostructural optimization examples

• Aerostructural optimization simultaneously optimizes the aerodynamic shape and structural properties of an aircraft to improve overall performance
• Considers the coupling between aerodynamics and structures, accounting for the loads, deformations, and aeroelastic effects
• Examples include the optimization of wing box structure, composite layup, and wing-fuselage integration
• High-fidelity CFD and FEA simulations are coupled to accurately predict the aerostructural behavior of the aircraft
• MDO architectures, such as collaborative optimization and analytical target cascading, are used to manage the disciplinary coupling and computational complexity

Computational tools for MDO

• MDO relies on a wide range of computational tools and frameworks to enable the efficient analysis, optimization, and integration of multidisciplinary systems
• These tools address the challenges of high-dimensional design spaces, complex disciplinary analyses, and the need for automation and collaboration

Optimization algorithms and solvers

• Gradient-based optimization algorithms, such as sequential quadratic programming (SQP) and interior point methods, are commonly used in MDO for their efficiency and convergence properties
• Heuristic and evolutionary algorithms, such as genetic algorithms and particle swarm optimization, are employed for global exploration and handling of discrete or non-smooth design spaces
• Optimization solvers, such as SNOPT, IPOPT, and NLPQLP, implement these algorithms and provide efficient solutions to large-scale nonlinear optimization problems
• Surrogate-based optimization techniques, such as efficient global optimization (EGO) and surrogate management framework (SMF), leverage surrogate models to guide the search for optimal designs

Integration of disciplinary analysis codes

• MDO requires the integration of disciplinary analysis codes from different domains, such as CFD, FEA, and propulsion simulations
• Multidisciplinary analysis (MDA) frameworks, such as OpenMDAO and ModeFRONTIER, provide a platform for coupling and executing disciplinary codes in a unified environment
• MDA frameworks handle the data exchange, unit conversions, and consistency management between disciplines
• High-performance computing (HPC) resources, such as parallel computing and cloud computing, are leveraged to accelerate the execution of computationally intensive analyses

Workflow management and automation

• MDO workflows involve the orchestration of multiple disciplinary analyses, optimization tasks, and data processing steps
• Workflow management tools, such as Pegasus and Taverna, enable the definition, execution, and monitoring of complex MDO workflows
• Automation frameworks, such as iSIGHT and ModelCenter, provide a graphical user interface for constructing and executing MDO workflows
• These tools facilitate the integration of disciplinary codes, optimization algorithms, and post-processing routines, streamlining the MDO process and reducing manual intervention

Future trends and challenges in MDO

• MDO continues to evolve and expand its capabilities to address the growing complexity and demands of modern engineering systems
• Future trends and challenges in MDO include the handling of high-dimensional design spaces, the development of efficient global optimization strategies, and the incorporation of machine learning techniques

High-dimensional design spaces

• As the complexity of engineering systems increases, the number of design variables and constraints in MDO problems grows exponentially
• High-dimensional design spaces pose challenges for optimization algorithms, surrogate modeling, and sensitivity analysis
• Techniques such as dimensionality reduction, feature selection, and adaptive sampling are being developed to efficiently explore and optimize high-dimensional spaces
• Advances in high-performance computing and parallel processing are enabling the handling of larger-scale MDO problems

Efficient global optimization strategies

• Global optimization aims to find the best solution in the presence of multiple local optima and nonconvex design spaces
• Efficient global optimization strategies are needed to balance exploration and exploitation, avoiding premature convergence to suboptimal solutions
• Surrogate-based optimization techniques, such as Bayesian optimization and kriging-based methods, are being developed to guide the search for global optima
• Hybrid optimization approaches, combining global search algorithms with local refinement techniques, are being explored to improve the efficiency and robustness of global optimization in MDO

Incorporation of machine learning techniques

• Machine learning techniques, such as deep learning and reinforcement learning, are being integrated into MDO to enhance the efficiency and effectiveness of the optimization process
• Deep learning models, such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs), are being used as surrogate models to capture complex system behaviors and reduce the computational burden of high-fidelity simulations
• Reinforcement learning algorithms, such as Q-learning and policy gradient methods, are being applied to learn optimal design strategies and adapt to changing problem formulations
• Transfer learning and multi-task learning approaches are being investigated to leverage knowledge from related design problems and improve the generalization of MDO solutions